Korhonen and Rybicki [1]} then showed that the \(\tilde{\Omega }(n^{1/2})\)  lower bound for 4-cycles can be extended and applies to \(p\) -cycles for all even values of \(p\) . Notably, in the spirit of the results of [2]}, it is shown in Censor-Hillel, Fischer, Gonen, Oshman, Le Gall, and Leitersdorf [3]} that going above this lower bound for \(p=6\)  would imply new lower bounds in circuit complexity, which are considered hard to obtain. The reason that this barrier for lower bounds works in the \(\mathsf {CONGEST}\)  model is because it is shown that it is sufficient to consider high-conductance clusters, and that these can simulate circuits in a similar manner to the \(\mathsf {CLIQUE}\)  model.
